Significant Figures

Our ability to measure a physical quantity accurately is fundamentally limited by the least count of the instrument being used — the least count is simply the smallest measurement that a given instrument is capable of resolving. For example, with an ordinary metre scale, the smallest division one can reliably read is $0.1\ \text{cm}$, so its least count is $0.1\ \text{cm}$; any digit beyond that in a reading is not truly known with certainty.

Suppose the length of a metal rod is measured three times with a metre scale of least count $0.1\ \text{cm}$, giving readings $15.4\ \text{cm}$, $15.4\ \text{cm}$, and $15.5\ \text{cm}$. The arithmetic mean (the most probable length, by the discussion in section 1.8.1) works out to $15.43\ \text{cm}$. Of these digits, we are fully certain about the '1' and the first '5' (the whole-number part), but not truly certain about the last two digits, precisely because of the least-count limitation of the instrument used.

This leads to the definition of significant figures: the number of digits in a measurement about which we are certain, plus one additional digit — the first digit about which we are not fully certain — together make up the significant figures (or significant digits) of that measurement. In the rod example above, we therefore report 3 significant figures: 1, 5, and 4 (i.e. we would properly report the length as $15.4\ \text{cm}$, not the full unrounded $15.43\ \text{cm}$). In general, the larger the number of significant figures a measurement carries, the greater the accuracy of that measurement — using an instrument with a smaller least count increases the number of significant digits obtainable.

Rules for counting significant figures:

1. All non-zero digits are significant. For example, a volume of $178.43\ \text{cm}^3$ has five significant digits: 1, 7, 8, 4, 3.

2. All zeros lying between two non-zero digits are significant. For example, $m = 165.02\ \text{g}$ has 5 significant digits.

3. If a number is less than 1, any zero(s) to the right of the decimal point and to the left of the first non-zero digit are not significant — they exist only to fix the position of the decimal point. For example, in $0.001405$, the three leading zeros are not significant, so this number has only 4 significant digits (1, 4, 0, 5).

4. Zeros on the right-hand side of the last non-zero digit are significant, provided the number is written with an explicit decimal point. For example, both $1.500$ and $0.01500$ have 4 significant figures each. (By contrast, a bare integer like $L = 125\ \text{m}$, if simply re-expressed with more zeros as $12500\ \text{cm}$ or $125000\ \text{mm}$ without any decimal point, still only carries 3 significant digits — the trailing zeros introduced purely by the unit conversion are not automatically significant.) …